Find the Other Trig Values in Quadrant II cos(x)=-2/5

Problem

cos(x)=−2/5, Quadrant II

Solution

1. Identify the given information and the signs of the trigonometric functions in Quadrant II. In Quadrant II, sin(x) and csc(x) are positive, while cos(x) sec(x) tan(x) and cot(x) are negative.

2. Use the Pythagorean identity sin2(x)+cos2(x)=1 to find sin(x)

sin2(x)+(−2/5)2=1

sin2(x)+4/25=1

sin2(x)=21/25

sin(x)=√(,21/25)

sin(x)=√(,21)/5

3. Find the reciprocal functions sec(x) and csc(x) by inverting cos(x) and sin(x)

sec(x)=1/cos(x)=−5/2

csc(x)=1/sin(x)=5/√(,21)=(5√(,21))/21

4. Calculate the tangent using the identity tan(x)=sin(x)/cos(x)

tan(x)=√(,21)/5/(−2/5)

tan(x)=−√(,21)/2

5. Calculate the cotangent by inverting the tangent.

cot(x)=1/tan(x)=−2/√(,21)=−(2√(,21))/21

Final Answer

sin(x)=√(,21)/5,tan(x)=−√(,21)/2,csc(x)=(5√(,21))/21,sec(x)=−5/2,cot(x)=−(2√(,21))/21

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